Transcript Solve for x.

CCGPS Analytic Geometry
GEOMETRY!!!
5 Ways to Prove Triangles
Congruent
1. SSS: All 3 sides are exactly the same
2. SAS: 2 congruent sides and the angle
in between
3. ASA: 2 congruent angles are the side
in between
4. AAS: 2 congruent angles and a side
NOT in between
5. HL: ONLY FOR RIGHT TRIANGLES –
Hypotenuse and 1 Leg
CONGRUENCE STATEMENT
Order matters!
Match up corresponding parts.
Example: ABC  DEF
Triangle Sum
The 3 angles in a triangle add
180
up and equal ______.
Exterior Angle Theorem
The 2 remote interior angles add up and
equal the exterior angle
Remote
Angle
Exterior
Angle
Remote
Angle
Isosceles Triangle
• 2 congruent sides
• Opposite of the congruent sides
are congruent angles
Rigid Motion – the shape will still
be congruent after the move
1. Reflection
2. Translation
3. Rotation
Dilate the figure by 1/2. Use the origin as the
center of dilation.
A  4,4  
A '  2,2 
B  2, 6  
B '  1, 3 
C  6,0  
C '  3,0 
Dilate the figure by 2. Use (-2,0) as the origin
as the center of dilation.
To do this, you have to
calculate the distance
each point is away from
the center of dilation
and then multiply that
distance by the dilation
factor.
A  0,0   A '  2,0 
B  0,3  
C  2,3  
D  2,0  
B '  2,6 
C '  6,6 
D '  6,0 
Find the center of dilation
Center
 2,2 
Similar Polygons
1. Corresponding angles are
congruent
2. Corresponding sides are
proportional
3. Similarity Statement
ABC ~ DEF
Solve for x and y.
ABC ~ SLT
L
A
10
cm
B
x
24 cm
x = 26 cm
y
5 cm
S
13 cm
C
T
y = 12 cm
In similar triangles, angles are congruent
and sides are proportional
ABC ~ SLT
Find the missing angle measures.
A
L
53
S
B
C
37
T
mC  37 mL  90
mS  53
Find the perimeter of the smaller
triangle.
12 cm
Perimeter = 60 cm
4 cm
Perimeter = x
x = 20 cm
3 ways to Prove Triangles Similar
1)Angle-Angle (AA~) Similarity
Postulate
2)Side-Side-Side (SSS~) Similarity
Theroem
3)Side-Angle-Side (SAS~) Similarity
Thm
Determine whether the triangles are similar. If so,
tell which similarity test is used and complete the
statement.
68°
43°
68°
43°
V
Y
7
W
11
3
U
X
5
Z
Prove that RST ~ PSQ
1. Two sides are proportional
SAS~2. Included angle is congruent
S
4
P
12
R
5
16 20

4
5
4 4

1 1
Q
15
T
S  S
reflexive
A tree cast a shadow 18 feet long. At the
same time a person who is 6 feet tall cast a
shadow 4 feet long. How tall is the tree?
tree's shadow
tree's height

person's shadow person's height
18 x

4 6
x  27
Trig Ratios
Trig Ratio
What is cos R?
What is sin R?
What is tan R?
21
29
20
29
20
21
Co-Function Relationships
sin   cos(90  )
cos   sin(90  )
1
tan  
tan(90  )
Co-Function Relationships
26
Cos 64 = Sin ____
Find a Missing Side
Solve for x. Round to the nearest tenth.
x = 17.6
x
Find a Missing Angle
Solve for . Round to the nearest tenth.

 = 31.4
The angle of elevation from a ship to the
top of a 35 meter lighthouse on the coast
measures 26. How far from the coast is the
ship? Round to the nearest tenth.
tan 26 = 35/x
x = 71.8 m
Angle Formulas to KNOW for the Test
Central Angle  Arc
Arc
AngleVertexOn 
2
Arc  Arc
AngleVertexINside 
2
Large Arc  Small Arc
AngleVertexOUTside 
2
Solve for x.
arc
76 
2
360  152
x  208
x
Solve for x.
110
40
x
110  x
40 
2
x  30
Solve for x.
38
A
D
B
x
C
148
38  148
x
2
x  93
solve for x
x  42
A
42
C
B
x
D
solve for x.
22  164
 93
2
180  93  x

x  87
S
22
T
C
A
x
Solve for x.
(Circle A)
A
x
168
168
x
2

x  84
solve for x.
x
120  110
x
2

x5
130
Solve for x and y.
x  98


y  39
Area & Circumference
Area   r 2
arc 2
r
Area Sector 
360
Circumference   d or Circumference  2 r
arc
2 r
Arc Length 
360
Find the arc length and area
of the shaded sector.
120
AL 
2 4.5
360
Arc Length  9.4 in
Asector
Asector
120
2

 4.5
360
2
 21.2 in
Formulas to KNOW for the Test - Segments
Part  Part  Part  Part
Outside  Whole  Outside  Whole
Solve for x.
2
x
3x
6
x2
Solve for x.
x
4
5
x  14.75
10
Question 18: Solve for x.
x
9
7
x  12
solve for x.
10
x
3
x4
5
Find the perimeter of the polygon.
9 cm


16 cm

8 cm
6 cm
P  16  16  6  6  8  8  9  9
P  78
Volume of Solids
Prisms/Cylinders
Cones/Pyramids
Spheres
V = Bh
1
V  Bh
3
4 3
V  r
3
B stands for the area of the base.
The shape of the base can change.
Area of Base
Circle =
2
r
Square/Rectangle = bh
Triangle = ½ bh
Trapezoid = ½ (b1 + b2)h